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Theorem undir 3722
Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
undir  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)

Proof of Theorem undir
StepHypRef Expression
1 undi 3720 . 2  |-  ( C  u.  ( A  i^i  B ) )  =  ( ( C  u.  A
)  i^i  ( C  u.  B ) )
2 uncom 3610 . 2  |-  ( ( A  i^i  B )  u.  C )  =  ( C  u.  ( A  i^i  B ) )
3 uncom 3610 . . 3  |-  ( A  u.  C )  =  ( C  u.  A
)
4 uncom 3610 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
53, 4ineq12i 3662 . 2  |-  ( ( A  u.  C )  i^i  ( B  u.  C ) )  =  ( ( C  u.  A )  i^i  ( C  u.  B )
)
61, 2, 53eqtr4i 2461 1  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    u. cun 3434    i^i cin 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-in 3443
This theorem is referenced by:  undif1  3872  dfif4  3926  dfif5  3927  bwth  20423
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